Laplace | MATHalino reviewer about Laplace

Problem 03
Find the value of $\displaystyle \int_0^\infty te^{-3t} \sin t ~ dt$

Problem 02
Find the value of $\displaystyle \int_0^\infty \dfrac{\sin t ~dt}{t}$.

Problem 01
Evaluate $\displaystyle \int_0^\infty \dfrac{e^{-3t} - e^{-6t}}{t} ~ dt$

If $F(s) = \mathcal{L}\left\{ f(t) \right\}$, then $\displaystyle \int_0^\infty e^{-st} f(t) \, dt = F(s)$.

Taking the limit as $s \to 0$, then $\displaystyle \int_0^\infty f(t) \, dt = F(0)$ assuming the integral to be convergent.

Theorem
If $\mathcal{L} \left\{ f(t) \right\} = F(s)$, then

$\displaystyle \mathcal{L} \left[ \int_0^t f(u) \, du \right] = \dfrac{F(s)}{s}$

For first-order derivative:
$\mathcal{L} \left\{ f'(t) \right\} = s \, \mathcal{L} \left\{ f(t) \right\} - f(0)$

For second-order derivative:
$\mathcal{L} \left\{ f''(t) \right\} = s^2 \mathcal{L} \left\{ f(t) \right\} - s \, f(0) - f'(0)$

For third-order derivative:
$\mathcal{L} \left\{ f'''(t) \right\} = s^3 \mathcal{L} \left\{ f(t) \right\} - s^2 f(0) - s \, f'(0) - f''(0)$

For n^th order derivative:

$\mathcal{L} \left\{ f^n(t) \right\} = s^n \mathcal{L} \left\{ f(t) \right\} - s^{n - 1} f(0) - s^{n - 2} \, f'(0) - \dots - f^{n - 1}(0)$

Division by $t$
If $\mathcal{L} \left\{ f(t) \right\} = F(s)$, then,

$\displaystyle \mathcal{L} \left\{ \dfrac{f(t)}{t} \right\} = \int_s^\infty F(u) \, du$

provided $\displaystyle \lim_{t \rightarrow 0} \left[ \dfrac{f(t)}{t} \right]$ exists.

Change of Scale Property
If $\mathcal{L} \left\{ f(t) \right\} = F(s)$, then,

$\mathcal{L} \left\{ f(at) \right\} = \dfrac{1}{a} F \left( \dfrac{s}{a} \right)$

Second Shifting Property
If $\mathcal{L} \left\{ f(t) \right\} = F(s)$, and $g(t)
= \begin{cases} f(t - a) & t \gt a \\ 0 & t \lt a \end{cases}$

then,

$\mathcal{L} \left\{ g(t) \right\} = e^{-as} F(s)$